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Definition df-trkg 25565
Description: Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence, (𝑥 𝑦) = (𝑧 𝑤) where = (dist‘𝑊)
  • for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊)
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2840.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (DimTarskiG≥‘2)
  • A9 Upper dimension axiom (V ∖ (DimTarskiG≥‘3))
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms DimTarskiG
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

Assertion
Ref Expression
df-trkg TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Distinct variable group:   𝑓,𝑝,𝑖,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkg
StepHypRef Expression
1 cstrkg 25542 . 2 class TarskiG
2 cstrkgc 25543 . . . 4 class TarskiGC
3 cstrkgb 25544 . . . 4 class TarskiGB
42, 3cin 3768 . . 3 class (TarskiGC ∩ TarskiGB)
5 cstrkgcb 25545 . . . 4 class TarskiGCB
6 vf . . . . . . . . . 10 setvar 𝑓
76cv 1636 . . . . . . . . 9 class 𝑓
8 clng 25549 . . . . . . . . 9 class LineG
97, 8cfv 6097 . . . . . . . 8 class (LineG‘𝑓)
10 vx . . . . . . . . 9 setvar 𝑥
11 vy . . . . . . . . 9 setvar 𝑦
12 vp . . . . . . . . . 10 setvar 𝑝
1312cv 1636 . . . . . . . . 9 class 𝑝
1410cv 1636 . . . . . . . . . . 11 class 𝑥
1514csn 4370 . . . . . . . . . 10 class {𝑥}
1613, 15cdif 3766 . . . . . . . . 9 class (𝑝 ∖ {𝑥})
17 vz . . . . . . . . . . . . 13 setvar 𝑧
1817cv 1636 . . . . . . . . . . . 12 class 𝑧
1911cv 1636 . . . . . . . . . . . . 13 class 𝑦
20 vi . . . . . . . . . . . . . 14 setvar 𝑖
2120cv 1636 . . . . . . . . . . . . 13 class 𝑖
2214, 19, 21co 6870 . . . . . . . . . . . 12 class (𝑥𝑖𝑦)
2318, 22wcel 2156 . . . . . . . . . . 11 wff 𝑧 ∈ (𝑥𝑖𝑦)
2418, 19, 21co 6870 . . . . . . . . . . . 12 class (𝑧𝑖𝑦)
2514, 24wcel 2156 . . . . . . . . . . 11 wff 𝑥 ∈ (𝑧𝑖𝑦)
2614, 18, 21co 6870 . . . . . . . . . . . 12 class (𝑥𝑖𝑧)
2719, 26wcel 2156 . . . . . . . . . . 11 wff 𝑦 ∈ (𝑥𝑖𝑧)
2823, 25, 27w3o 1099 . . . . . . . . . 10 wff (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))
2928, 17, 13crab 3100 . . . . . . . . 9 class {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}
3010, 11, 13, 16, 29cmpt2 6872 . . . . . . . 8 class (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
319, 30wceq 1637 . . . . . . 7 wff (LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
32 citv 25548 . . . . . . . 8 class Itv
337, 32cfv 6097 . . . . . . 7 class (Itv‘𝑓)
3431, 20, 33wsbc 3633 . . . . . 6 wff [(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
35 cbs 16064 . . . . . . 7 class Base
367, 35cfv 6097 . . . . . 6 class (Base‘𝑓)
3734, 12, 36wsbc 3633 . . . . 5 wff [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
3837, 6cab 2792 . . . 4 class {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
395, 38cin 3768 . . 3 class (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
404, 39cin 3768 . 2 class ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
411, 40wceq 1637 1 wff TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Colors of variables: wff setvar class
This definition is referenced by:  axtgcgrrflx  25574  axtgcgrid  25575  axtgsegcon  25576  axtg5seg  25577  axtgbtwnid  25578  axtgpasch  25579  axtgcont1  25580  tglng  25654  f1otrg  25964  eengtrkg  26078
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